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Is there a deterministic guide to landing?
Much of the aviation related instructions seem clear-cut and science-based.
For some reason landing seems heavy on art and light on science. To explain, I’m told to round out at about “yea” height, which does not register on any instrument. Even Rod Machedo calls it "the Art of Landing".
Depending on how you feel that day about the runway fixed point rushing at your face at 67 kts, “yea-15ft” could mean surprise wheels touching down (before solo), or stalling high in ground effect and plopping onto the runway (my 2nd solo).
Understanding the math in Jacobson’s flare and Rod’s expansion rate,  Lindberg-reference, it seems frustratingly unstructured learning when it comes to landing.
Setting the “it’s something you get the feel for” advice aside, is there a better way of learning landing that I’m missing? Really appreciate you help here.
Edit

The circuit, approach, and glide-path are fairly repeatable accurately using feedback from the instrument cluster even in variable conditions.
The goal here is to establish a method to allow consistency from pilot to pilot without relying on seat-of-the-pants black-box terms such as human skill, feel, art, etc.

Edit 2
Explaining the issue further
Having seen the responses from community, it appears that we're simply asking PPLs gray-matter to act as the Height Above Terrain instrument. And everyone's OK with this despite the fact we're well trained about looking out visual illusions and [not] trusting the grey-matter in IMC.
If we're honest with ourselves, we should be able to acknowledge the absurdity in this even though it's been fine for 100+ years.
Allow for some humour: The landing skill as it's currently described looks suspiciously like practicing and getting good at guessing how many jellybeans are in a jar with the winnings being your existing nose-gear.
One can feel the stall onset a good bit before the stall-horn goes off or the actual stall happens, but we're not asking the pilots' brain to act as a stall-horn.
Yes, we can train the brain: If you've done any inversion training on the trampoline you'll notice how quickly "air sense" develops and know your exact position and attitude during a flip [the skill].  Yet, we all watch highly trained olympic athletes bumble it, and be inconsistent
 The kind of guide/indication the question is seeking
If you apply the 1/60 rule during landing you can then conclude that for every ~20ft of forward travel (at 3-degrees glide-path), you're dropping 1ft in height.
This can be very helpful in knowing when and were to round out and Jacobson's Flare seems based on this.  Furthermore, the paint marks on the runway are known lengths so you can tell height loss passing the start and end of a painted segment.
The trouble is: you need to have a known starting height e.g. height at threshold; and the Altimeter, or one's brain aren't the most accurate of tools for this.
The question is investigating the possibility of other visual references queues that could inform the pilot regarding ground-effect and round-out height similar to the not-so-easy-to-miss sign of buffeting before some stalls.

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Landing an airplane is neither art nor science, it is a skill.
Like any skill requiring hand-eye coordination, muscle memory, reflexes, timing and judgement, it requires practice.  It doesn’t matter how much you know about the subject or how you feel about it, the only way to get better is to actually do it.
For example, consider the act of hitting a baseball:

You can use science to calculate the speed of the pitch, the timing of the swing, the power and the trajectory needed to clear the fence.
You can use art to describe the grace of a perfect swing, the crack of the bat, the roar of the
crowd, and the feeling of elation from rounding the bases after
hitting a home run.

But the only thing… the ONLY thing that gives the batter a chance of hitting that 95 mph fastball is years of coaching and countless hours of batting practice.
I don’t know what kind of “deterministic guide” you think might actually be useful, but to answer your question, there might be...  A few respondents here have provided some useful information that perhaps might satisfy you.
But you cannot intellectualize yourself into being a better pilot, the only way improve is through repetition.
ADDENDUM:
First let me acknowledge that I haven’t answered the question as it is worded.  However, as I have tried to make clear above, landing an airplane is a psychomotor skill that requires a blend of cognitive functions with physical movement.  As such, you cannot separate one from the other and hope to achieve success.
In that regard I believe the question is flawed in that it would appear to be asking for a purely cognitive approach.  OK, maybe not purely cognitive since the pilot trainee would at some point need to engage in the physical act of moving the controls, but it (apparently) seeks to sever the very feedback loop that helps program the brain to duplicate the motions resulting in a “good” landing while avoiding those that result in a “bad” landing.  (i.e. the emphasis in the question is on NOT relying on seat-of-the-pants “feel”, skill, etc.)  Our brains are hardwired to process sensory input, so to willfully disregard this tremendous benefit seems ludicrous.
While the cognitive aspect is important, it cannot stand on it's own.  Improvement can only occur when it is paired with feedback from physically performing the action.  The exception to this is that once the brain has a sufficient database of good and bad feedback it is possible to mentally rehearse and improve through visualization.  (i.e. "chair flying")  And to further the importance of practice over intellectual cognition, consider the human infant:  Knowing absolutely nothing about the incredibly complex equations of motion required to program a robot to balance and walk, they are somehow able, (by physically performing the action and processing the "feel" as feedback...) to learn to balance on their feet and walk.
My personal technique while instructing was to actually emphasize sensory feedback in an attempt to instill a Zen-like oneness with the machine; the airplane as extension of the body... (For example:  "Did you sense that sinking feeling right before you noticed with your eyes that we were low?  Do you remember how far back you had pulled the power just before that?  Do you think you may have overcorrected for the high/fast state?  Do you remember the feeling of how much less you corrected for the previous one where you nailed it?  The sound of the engine, the relative position of your hand? Yes?  Good, then lock it into your brainpan and try to do that again!")
Finally, flight instructors need to be aware of each student’s unique learning style and adapt their teaching as much as possible to optimize results.  If the goal is consistency from pilot to pilot a single source deterministic guide runs counter to that principle.

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Yes indeed, reading Height Above Terrain with a resolution of a couple of feet is not accommodated by the classic aircraft instrumentation set, particularly not the set in aircraft used for initial training.
Learning how to fly is for a large part training the body to control the visual and motion sensory inputs. Not every aeroplane has a HAT instrument, so it is not a bad idea to get trained in making an assessment just based on out-the-window view and the feel of ground effect kicking in.
Reading an instrument takes more time than feeling and seeing motion, and time is critical when touching down. And there is the need for the human to teach the body how to land, pick up landing skills as mentioned in @MichaelHall's answer.
Indeed, autoland touches down algorithmically, with higher repeatability than when humans land. Basis of that are transmitters of signals that we humans cannot pick up, the receivers in the plane can. And now, what we do not want is that the plane can only touch down when the signal receivers are working properly. As ever, in civil aviation the three most pressing factors are safety, safety and safety:

Civil aviation is super-safe, but of the relatively few accidents the highest proportion is in landing. From this link:

Nearly half of all worldwide commercial jet airplane accidents during the 10-year period from 2004 through 2013 occurred during the final approach or landing phases of flight, according to statistics published in September by Boeing Commercial Airplanes.

Fully automated flights with no pilots on board are not perceived well with paying passengers.
And humans must be able to take over in emergencies, as mentioned in the answers to this question. Captain Sully in the Hudson indeed.

Yes there are algorithms that can land an aeroplane deterministically, but well experienced skilled humans must be present to take over in an emergency and get us safely on the ground when the avionics systems are not in order.

That's why aircraft that do have radar altimeter accurate to a feet announce the height with synthetic voice.

And when your RA or speakers are broken, you make the PM give you callouts based on baro alt.

And when you don’t have a pm?

In general, flight instructors are not going to get this detailed with you for the practical training and tend to just concentrate on what you need to do in landing a particular airplane, but I will offer this analysis for what it’s worth.
Officially, landing is the means by which an aircraft transitions from flying to being supported on the ground by its landing gear.  The techniques to do so vary from aircraft type to aircraft type, but all typically involve the aircraft making a very gentle descent towards the ground, thence contacting the ground, mooring mast, etc. on its landing gear and remaining there with the weight of the aircraft transferred from the wings, rotors, or other lifting surfaces, to the landing gear to then taxi to a parking area, shut down, etc.
For airplanes, gliders, and other fixed wing aircraft, this process typically involves two stages: an approach to a runway followed by a roundout.  The approach segment involves flying the aircraft along a specific path through space guided either by visual reference to the ground or electronic instruments in the case of an instrument approach. Approaches can be straight in, circling, or involve a traffic pattern. The goal of any approach is to line the airplane up with the runway centerline, along with a vertical guidance, at the minimum airspeed needed to enter the roundout phase.  This speed is typically 1.3*Vso or approx ~30% faster than the stalling speed of the airplane in the landing configuration.  Approaches will either terminate in a roundout, a go-around for a traffic pattern, or a missed approach procedure in the case of an instrument approach.  Go-Arounds and Missed Approach Procedures involve terminating the rate of descent to the airport or runway, applying power, and configuring the aircraft for a maximum rate of climb to either re-enter a traffic pattern or fly an instrument course to set up for another instrument approach.
If the termination of the approach and a go-around or missed approach is not needed, the next phase will be the roundout.  The goals of the roundout (sometimes called flare) are:

Arrest the rate of descent of the aircraft toward the runway about 1-2 feet above the runway
Place the aircraft into the correct attitude so it
Makes the appropriate contact, aligned properly with the runway,  on its landing gear
Accomplishes all this while covering the minimum amount of runway without excessive ‘floating’

All non-VTOL fixed wing aircraft are landed in this manner.  The exact methods and techniques for doing so will vary between aircraft type to aircraft type and depending on the runway type, landing gear on the aircraft, etc.
That’s all great for a textbook, but human beings are three-dimensional creatures who experience life very differently than a textbook description.  Pilots are artists, because flying an airplane is a function of the right side of the brain.  We think about this problem spatially, taking visual and kinesthetic cues, which we can use to determine whether we are or are not performing this task correctly.   And a generic textbook description is not going to cover that.   How, for example, do you determine how high you are off the ground in the flare?  While propwash effects are a factor in landing a small trainer airplane, it is not a factor in flying a jet.  How do you anticipate the length of a flare so the aircraft touches down where you want it to?  What are the flight characteristics of the airplane when landing during gusty winds, or crosswind conditions?  The process of controlling the airplane itself is an artform.
Given all the variations that go into landing an airplane it’s difficult to create a deterministic models as you put it for getting good landings.  But there are some recommendations.

On a good VFR day in the traffic pattern, try to envision the traffic pattern itself, including all legs of the traffic pattern in the sky around your airport, then make the airplane fly that path. It sounds a little crazy but it does provide good guidance for where you need to be and visualize where you need to be in order to make the airplane do what you wanted to do.

Good landings all start with good traffic patterns, and good consistency is the key to good traffic patterns. One of the best ways to achieve consistency is knowing what power settings for what configurations of the aircraft you will need to achieve a certain rate of descent, airspeed, etc.

If you’re operating out of an airport where traffic is light, or if the tower will grant this at a towered airport, asked to fly a larger traffic pattern, and make notes on a knee board of the power settings, trim, flap configurations, etc. that you need to achieve stabilized flight at certain airspeed.  As an example, flying a C172 in the downwind, you not that the airplane, flaps up and trimmed just slightly nose down of the takeoff mark requires a power setting of 1900 RPM to maintain level flight at 85 KIAS.  Note these metrics for all legs of a traffic pattern and then fly them that way in the airplane. If you have trouble remembering those things on the go, take a session where you chair fly in front of a cockpit poster doing all the steps of a traffic pattern, setting power, configuration, trim, etc. until you have it memorized and it becomes second nature.  If you do this you will find the airplane is very controllable in all legs of a traffic pattern and any kind of adjustments, only require slight changes in power and/or trim.  I especially like practicing long final approaches this way.  This advice is indispensable when transitioning to a new airframe type.

Following the steps I described above, start practicing picturing the pattern as a transparent path in the sky and make the airplane fly alongp that path. You will find it reduces a lot of errors in position, altitude, and speed if you do it that way.

Base to final turns can be a struggle for students, often times causing the student to overshoot or undershoot the runway centerline.  To guard against this, on base, project the runway centerline out to a point where it intersects your base leg flight path and prepare to turn inside of that corner. You should line up right on final as you roll out if you do this right.

Timing the start of a round out and flying the round out can be more of an art than a science.  It’s a matter of hand-eye coordination combined with kinesthetic fuel for the aircraft.  Rod Machado’s runway expansion method works well for light trainers, but this is just something you gain from a lot of time in practice. If you’re making bad landings on the way you’re pretty normal.

Another exercise during dual instruction are high speed taxi flares.  Basically, you and the flight instructor will go out on a (long!) runway and accelerate the aircraft up to approximately 10 knots below its rotation speed, thence set power to hold this speed and apply back pressure on the yoke to lift the nosewheel off the runway, effectively doing a high-speed taxi wheelie down the runway.  This will give you practice with two different skills. The first is gaining an appreciation and kinesthetic feel and response of the flight controls in the roundout as well as the approximate sight picture you want to see in the roundout.  Note, if the airplane becomes airborne at this point, immediately ease off some back pressure and gently reduce the throttle while holding that same nose high attitude you would in the flare.  This should cause the airplane to touch down again.  Always plan on terminating the exercise with at least 2500 feet of paved, dry runway remaining so you can slow down to taxi speeds under positive control without risk of overshoot or loss of control.  NOTE WELL: This is an exercise to be performed on dual instruction flights only.  NEVER attempt this on a solo flight as a student pilot.

Still another exercise is to have the student fly a low approach 1-2 feet above the runway on Vref (approach speed).  This, too, will provide an excellent site picture in kinesthetic experience of the roundout.  If crosswind conditions exist, the student should be applying the correct cross control technique to keep the aircraft aligned on the centerline of the runway.  With approximately 3000 ft of runway remaining, a proper go-around technique should be initiated to allow the airplane to establish a best angle of climb to clear obstacle at the departure end of the runway.  I favor this technique with new students who have little or no aviation experience as the transition to fully landing the airplane.  All that’s required to land the plane in this low approach exercise, is to smoothly pull the throttle to idle while simultaneously holding the airplane off from touching down using back pressure.  NOTE WELL: This is an exercise to be performed on dual instruction flights only.  NEVER attempt this on a solo flight as a student pilot.

These techniques aren’t magic but they might just make it easier to flare and land by taking some of the mystery out of it.

thank you for the detailed post and the exercises.  It's a great answer for the human skill development part.

Autopilots regularly perform landings, so yes, landing can be described as an algorithm. But since human brains are unable to perform such an algorithm, we depend on our neurons to build pathways that approximate the "perfect" solution - these pathways we perceive as experience and gut-feeling.

As mentioned in other answers, computer programs exist that can land a plane. That shows that algorithms exist to land a plane. That doesn't mean that a human can mechanically implement said algorithms.
OP compares landing with parking, saying that they apply an algorithm to park. The difference with landing a plane is that, compared to parking, there many more variables to control. The pilot needs to control attitude (in particular if there is turbulence) horizontal speed,  vertical speed, centering the runway, gliding slope. And the flare, which requires reducing speed and putting the plane in the right attitude for touchdown. In addition, the plane's behaviour will vary depending on pressure altitude, plus one needs to take the wind into account; both the headwind part, that affects the glide, and the side wind, that affects the centering on the runway.
When pilots learn to fly, they are taught many techniques to deal with all the above. Eventually it becomes second nature, but it requires constant training to stay sharp. As opposed to parking, the consequences from a mistake are at a different level. And, also, different aircraft behave very differently, affecting all the above.

"You need to have a known starting height ... when and where to
roundout ...

You're overthinking this.  Whether you fly straight in from 10 miles away of roll out onto final from your baseleg, the key is proper glide angle to reach the runway.  Height is only important if there are obstacles to clear. You job on final is to fly a safe approach speed (trim for 67 knots), and adjust your glide angle with power/flaps as needed.
Because a little extra speed is carried, the "round" can be one or more steps down.  Round to 15 feet, ease down to 5 feet, pull power, land.  This takes practice.  In light aircraft, because of wind, landing cannot be done by rote.
When you get good, rounding will get better.
Roundout height will depend on your aircraft, but airspeed is most important during this phase of landing. Rounding out is a distinct and separate phase of landing, properly done at approach speed.
In a Cessna 172, a good round out is done at around 10-15 feet.  Remember, you are still flying the plane here, preparing for the flare and then the landing.  You now have time for any aileron and rudder adjustments to be in line with the runway.
Next allow speed to bleed off, gradually raise the nose. This is the "flare". Now you're landing.  Look at this as "the next step down".
If you have sufficient runway, a little "float" is fine, don't try to be perfect.  "Short field landings" come later, but a slower approach carries far greater risk in variable winds.  Stick with that 67 knots approach speed for now, and be consistent with it, not too fast either.
Pick the longest runway available at first and practice.  It gets better with experience.  Work on "setting it down" from your roundout height.  As speed decreases, the plane essentially lands itself.

“around 10-15 feet“.  At 500ft/min in the glide slop is less than 1 second margin of error.  No?

@MandoMando  that would be level to 10-15 feet (as a "step").  When you start to round depends on your glide angle.  This is why we all say practice, as a power on approach will have an angle of around 3 degrees, whereas power off full flaps can be around 5-6 degrees.  Do not fool around with slower "short field" approaches until you get a good feel of rounding.  Your second step down (as you are floating) will be to 3-5 feet as speed bleeds off and your nose comes up.

A small anecdote here. With a bit of analysis later.
I used to haul gliders up into the air mostly using a Piper Supercub. On a good day I did 20 to 40 starts, and landings of course. After a while it can become a bit boring so I invented small challenges for myself. Such as always leaving the glider in the best updraft around the field at at end of tow (the glider pilot does the disconnect). A bit of interest as the updrafts moved around during the day. Another challenge was to make 10 perfect landings in a row. Now what a perfect landing consists of can be argued about, but suffice to say I never managed to get above 3 in a row.
The old saying is that a good landing is any landing you can walk away from. A perfect landing, in my challenge, would be go well above that. It would include the exactly correct airspeed, flaring at the exactly correct height and touching down at the exactly correct place. Between landings the conditions changes: mostly wind speed and direction would change all the time. Literally one minute it could be mostly still, the next minute a thermal bubble would leave ground and induce a 10 knot tail wind. A perfect landing would cater for these differences.
In a very small and light plane as the Piper Supercub these environmental things, say a sudden tail wind has a great effect on how the plane lands. In a very large plane, 10 knots of wind makes much less of a difference of course. The old, almost antiquated, instrument set in the Supercub did not include any instruments helping in landing above the air speed indicator. And, well, the flying is "by the pants" anyway, after a while you hear and fell airspeed well enough to not really need any instruments in.
Knowing how to land without instruments (covered), was actually part of the curriculum when I learned flying gliders. Of course, they tend to be much easier to land than planes with a heavy iron in front. And I think that as long as we fly around in basically 1950 motor planes it is a good idea to practice - it can happen. Of course, in the far future, computers and instruments will take over this and no-one will be required to land manually.
So in effect, accept that landing is not a "analytical art", but a learned skill. Similar to learning how to ride a bike or to play an instrument, only a bit more involved. A good teacher can help but it is you that has to learn.


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What transatlantic flight routes are available for a Pilatus PC-24 with std configuration
I'm interested in this plane, but the aircraft's advertised range is 1,800nm with 6 passengers onboard.

So a direct flight is not possible.
How do you fly from e.g. Paris to New York? What are the possible routes for shortest time, or maximum safety margin.
What stops do other aircraft of similar category make?

Welcome to aviation.SE! This question or this one might be helpful if your question is about range. If it is, you might also ask yourself "how can I drive further than my car can go on a single tank of gas?". Or maybe you're asking "where can aircraft refuel when crossing the Atlantic?"; if so, feel free to clarify that.

wow I'm surprised by the downvotes guys! this doesn't seem a super welcoming community

@julien_c I suggest to edit your question to ask for specific suggestions such as routing, pilot experience, expected weather conditions and other route specific information. If you agree, I can edit it into your question.

Why do you think the answer would be different from the general case of flying any plane to any destination beyond its fuel range?

If your question is about routing, you would have to fly something like LFPB - LPAZ - CYYT - KTEB for instance, or substitute BIRK for LPAZ. Choose the best routing according to fuel cost, ease of obtaining overflight clearances and visas* if you are going to stop over, and handling costs. Weather will also be a consideration. I've done this plenty, get yourself a good handling company.
Make sure you are familiar with the radio reporting procedure, there is a very specific format and is found on the Jeppesen North Atlantic charts. If you have CPDLC all the better. There is also a very specific radio procedure for receiving oceanic clearance and readback.
Note: A route like this is not for a novice! You need to be familiar with Atlantic crossing procedures.
*That will be a very long flight, you might have to stop over at one of the intermediate stops. Santa Maria is very nice and if I am not mistaken, the Schengen visa covers it.

LFPB - LPAZ - CYYT - KTEB sounds good and shouldn't be very complicated visa-wise.
A friend of mine would be the pilot though he would still need to get his jet certification.
Thanks for your reply! I think we should be able to make it in ~24 hours one way including the stops, no?

As I said I would recommend that an experienced pilot be on board, one who knows the Atlantic crossing protocols and who has done this before. And make sure the weather forecast is good,

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Would atmosphere floating inside an airship be counted towards the mass of the airship?
Airships are filled with hydrogen.  However as pointed out by Peter Kampf in the answer to the question below when an airship is on the ground only a portion of its hydrogen cells are filled with hydrogen.
During WW1 Zeppelins began their missions with Hydrogen cells at 33% full. How did this provide enough lift to get off the ground?
This chart details the hydrogen density in a given airship before takeoff.

LZ104's cells are only filled to 42% at sea level.  Does this mean that below the hydrogen cells sits an empty volume of air equal to roughly 58% of the volume of the cells?  If so is this air counted towards the mass of the airship?
I ask because water in a ship's hull makes it heavy and sink.  Why wouldn't air in an airship make it heavy and fall to earth?  Or would it?

Water in a ship's hull displaces air which would help to let it float. If that water would displace something of equal density (cargo, for example) it would not make the ship heavier.

From a physics perspective all matter has mass. So yes, the air counts as mass.

Or in other words: if there's water in a ship's hull that means there isn't air in the ship's hull and if there's not enough air then it sinks. But some water is okay. In fact since ships have toilets, you know that it's okay for them to have water.

Yes towards inertial mass, no towards gravitational mass.

Or to make it more clear (I hope) the mass of the air doesn't count towards lift (because it's the same inside and out).  However, it contributes to inertia, since you have to accelerate that air along with the rest of the aircraft.

If that was true @ZsoltSzilagyi we'd have no atmosphere.

@GdD: That was a shorthand for "yes, it has a weight, but it is counteracted by the buoyancy".

Quick note: Modern Airships (Zeppelins) do not use Hydrogen any more, but Helium. This is heavier than Hydrogen, so the lift is smaller. But Hydrogen is just to dangerous, see the fate of LZ129.

Would atmosphere floating inside an airship be counted towards the mass of the airship?

Yes, but not for buoyancy.

Does this mean that below the hydrogen cells sits an empty volume of air equal to roughly 58% of the volume of the cells?

Yes, exactly.
When calculating the lift produced by the interior of the airship, the gas with lower density compared to outside air (hydrogen) produces lift while the internal air of equal density to outside air has no influence. It simply cancels out. While hydrogen displaces air of higher density, creating lift in the process, this internal air displaces outer air of equal density. So for lift that internal air can be disregarded.
However, when maneuvering this air does count. It increases the inertia of the ship, both when changing direction and when accelerating or decelerating. To this air the airship designers even counted the outside air which is influenced by the motion of the ship. An airship in motion continuously displaces air at its bow and lets it flow together again at the stern. The air is thus given an impulse which - as the product of mass and speed - can be described as an additional mass of the airship at this speed. In the case of an elongated body, this additional mass in the direction of travel is small in relation to the mass of the displaced air, but is of the same order of magnitude in case of lateral or vertical movements. In case of rotational movements, the inertial forces and moments of the gases within the airship envelope are added. To make it easier to calculate the forces on the airship hull, the concept of virtual mass was introduced. This virtual mass depends on temperature, altitude and speed. Although this technique allows a simple mathematical description of the effect near an operating point, it can only be generally applied with additional effort. If you are interested in the actual values: NACA-Report 184 contains tables and correction factors for the calculation of virtual mass.

I ask because water in a ship's hull makes it heavy and sink.

Because this water would displace air which is lighter than water and makes the ship float. This comparison only works when air would be replacing hydrogen in the gas bags. See the empty volume in the airship as akin to an external tank which is added to a ship and submerged. Its weight would be carried by the water. The air-filled volume in an airship is needed to have space for the gas bags to expand into without disturbing the aerodynamic shape of the hull.

Why wouldn't air in an airship make it heavy and fall to earth? Or would it?

Because this air displaces air of equal density. It cancels out in the buoyancy calculation.

Thanks Peter that is fascinating.  So when an airship has a large diameter it produces tremendous lift.  However the hull is also full of air which makes the ship more challenging to turn left or right.  Very interesting.

It all comes down to buoyancy.  One would actually be heavier if weighed in a vacuum chamber.  Water entering a ship does not make it sink, just as air in a dirigible does not affect its weight (if it is being weighed in an air medium).
What makes a ship sink is anything denser than water, like steel.  To float, the volume of steel must be lightened by adding a volume of something less dense than water (steel + air = water density).  The displacement of the ship is the amount of water volume that is of equal density (there for weight) of the entire ship (including superstructure).  If the ship floods, it's density becomes greater than water (steel + water > water density).
So it is with airships: hydrogen + frame + engines + fuel = density of air for a given volume (weight) of air they displace.  In an air medium, air in the ship does not count.
It may be noted that both airships and submarines can rise or sink by trading light volume for heavy (submarine air and water, air ship hydrogen and air), but lift can also be used when under power for the same purpose.

Fun fact:  an airliner flying at 35,000 feet would actually have to account for the higher density of pressurized air in the cabin as extra weight.

How would the airliner actually account for it?

@MichaelHall The density is easilly got from the equation of state rho = p / (R * T). The added mass density is only (p - p_outer) / (R * T). It is a similar problem to weighing the same object in air and in vacuum - sometimes the difference is negligible, sometimes it is not.

@RobertDiGiovanni  That's a fascinating point about pressurization.  So if somehow airship cabins were pressurized the extra air would have been counted towards mass.  I hadn't thought of that.

A380 pressurized to 8000ft would be ~1.2tonne heaver than with 35000ft density given 2100m^3 pressurized volume.  That's about 2x the mass of the fuel that is burned during takeoff and climb to 1500ft.

Airliners pressurise to 8000' because that's as low density as possible, to reduce mass, without starting to run the risk of medically embarrassing the passengers.

@Vladimir F, you missed my point.  I wasn't asking how to calculate density, I was asking how an airliner "accounts" for it...  (and I am being facetious)  ;)

@MichaelHall I think I did get thee point you want to ask about where it appears in the FMS or somewhere but it does not, at least not visibly. The mass of the plane is not actually bigger in the altitude. It is only less buoyant. That means it is one of the many aerodynamic and aerostatic forces that change with the aircraft position, speed and attitude and it is hiddent in the respective calculations. The mass displayed wherever does not change.

Dude, please look up the word facetious.  And understand what ;) means...

As the airship ascends, the air is displaced by the expanding hydrogen.
What this means practically is that LZ104 filled to 42% can fly at a higher altitude than LZ40 filled to 70%, because its H2 can expand from 42% to 100% (2.5x) rather than from 70% to 100% (about 1.5x) before it runs out of lift.
As you can see in the "max altitude" figures in that table, LZ104 can reach 8200m vs LZ40's 3900m.
An extreme example is those high altitude balloon launches where you can see a tiny bubble of helium at the top of a huge envelope.

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When an airship climbs the hydrogen in the ship's cells expand. Does this mean that when the airship descends the empty cells fill with air?
During WW1 a U Class Zeppelin had hydrogen cells that contained 55,795 cubic meters of hydrogen.
At sea level the cells were filled to 100% with hydrogen and this weighed roughly 4,500 Kilograms.  The cells displaced roughly 68,000 Kilograms of air and this difference gave the ship tremendous lift.
As the ship ascended to 5,000 meters air pressure would drop by 50% and the hydrogen would expand.  To keep the cells from popping the crew would release a significant amount of hydrogen.
However this is where I see a problem and I'm not sure how engineers solved it.
As the Zeppelin returned to sea level with hydrogen cells at only 50% full wouldn't it take on air in the now empty 50%?  The air is heavy and at some point the airship would come crashing down to Earth.  How did they solve this problem?

This Wikipedia article, while perhaps not answering your question directly, makes for an interesting read.

@Sanchises I find it especially noteworthy that it was destroyed half a year before it even  made its maiden voyage (see box on the right). I did't know that British defence was that effective.

The last paragraph is not correct. Bleeding hydrogen lowers lift. But while some air has to be let in, lift does not change during descend.

You do not let air in to the cells (especially if your lifting gas is hydrogen!).  Instead, the volume of the cells reduces.  For a practical demonstration, blow up an ordinary balloon and take it to the bottom of a swimming pool.

First of all, Zeppelins carried tons of water ballast in order to compensate for hydrogen seeping from the gas cells or being vented when the ship had climbed too high. Before launch, the crew calculated the ballast need based on the planned length of the flight and weather conditions. Also, the ship lost mass continuously by burning off fuel. Below you see the calculation sheet of LZ 40 from May 1915. The ship took 7.5 tons of water, distributed over several tanks along its length.

The top row of the sheet lists fuel (Benzin, 9320 kg in total), then oil (520 kg), then a side view of the ship with its 16 gas cells and below that the water ballast. The crew weighed 334 kg.
On the ground the gas bags were never full. In the later WW I Zeppelin models, only one third of the internal volume was taken up by the gas bags and their hydrogen while the rest of the hull was filled with air. As the airship climbed, the hydrogen in its cells expanded and pushed the air out. On the way down the reverse happened. The gas bags were flexible and sufficiently airtight so no air would enter them.
The photo below is from airships.net and shows the LZ-129 Hindenburg under construction. The gas cell is partially filled and is attached to the circular frames over two thirds of their circumference to distribute the lift of a partially filled cell as evenly as possible. Wire bracing distributed the load over the full frame. In the center of the hull you can see a walkway for inspection; the gas cell had a central tunnel to allow lengthwise movement over the full length of the ship. The vertical funnels for venting leaked hydrogen are not yet installed.

This is only possible with rigid ships; blimps use ballonets for the same purpose but have to pressurize them a bit in order to keep their envelope taut.

Fascinating answer Peter!  I learned so much.  If it's alright I'm going to ask a follow up question.  I'll make it a new question because I'm not sure if there is enough room in comments.

@myNewAccount: Yes, go ahead. That is the right way to do here!

Addendeum here, https://aviation.stackexchange.com/questions/81594/during-ww1-zeppelins-began-their-missions-with-hydrogen-cells-at-33-full-how-d


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During WW1 Zeppelins began their missions with Hydrogen cells at 33% full. How did this provide enough lift to get off the ground?
During WW1 U Class Zeppelins had hydrogen cells that contained 55,795 cubic meters of gas.

According to Peter Kämpf's excellent, detailed explanation of Zeppelin mechanics they began missions with their hydrogen cells filled with roughly 33% hydrogen.  The rest of the volume contained air.  Because hydrogen expands as air pressure falls starting at 33% allowed Zeppelins to reach high altitudes without rupturing their gas cells.
However given the constants below I'm not sure how 33% hydrogen produced enough lift to get a Zeppelin off the ground.
Sea Level, 0 Meters:
Temp Celsius: 15°
Temp Kelvin: 288.15°
Pressure Pascals: 101,325
Gas Moles per cubic meter: 42.29
Air mass per cubic meter: 1.221 Kg
Hydrogen mass per cubic meter: .085 Kg
55,795 cubic meters of gas cells * 33% full = 18,412 cubic meters hydrogen
18,412 cubic meters of displaced air mass = 22,481 Kg
18,412 cubic meters of hydrogen = 1,566 Kg
Useful lift = 22,481 kg - 1,566 kg = 20,915 Kg
The empty weight of a U class Zeppelin was 25,750 Kg.  That's without fuel, ballast or bombs.  If useful lift was less than empty weight how did it take off?
At 5,000 meters altitude useful lift stays the same because although the hydrogen cells expand and displace more air the atmosphere is less dense and has less mass per cubic meter.  Lift at 0 meters altitude is the same as lift at 5,000 meters altitude.
Another detail I found interesting is that according to Wiki the U Class Zeppelin produced 64,750 Kg of useful lift.
When I plug in numbers at sea level that suggests that the gas cells were filled to the top with 100% hydrogen.
55,795 cubic meters of gas cells
55,795 cubic meters of displaced air mass = 68,126 Kg
55,795 cubic meters of hydrogen = 4,743 Kg
Useful lift = 68,126 Kg - 4,743 Kg = 63,383 Kg
However if the hydrogen cells were filled to 100% it couldn't climb without bleeding hydrogen all the way up.  It should be noted that at 5,000 meters the air is thin and useful lift drops by almost half.
Altitude, 5,000 Meters:
Temp Celsius: -17.5°
Temp Kelvin: 255.65°
Pressure Pascals: 49,586
Gas Moles per cubic meter: 23.33
Air mass per cubic meter: .674 Kg
Hydrogen mass per cubic meter: .047 Kg
55,795 cubic meters of gas cells at 100% full
55,795 cubic meters of displaced air mass = 37,605 Kg
55,795 cubic meters of hydrogen = 2,622 Kg
Useful lift = 37,605 Kg - 2,622 Kg = 34,983 Kg
34,983 Kg of lift is maybe barely enough to stay afloat with fuel, ballast and bombs.
But even if they did it this way when the Zeppelin descended the Hydrogen cells would be mostly empty and would fill with air causing the ship to gain weight and crash.
At 0 meters
55,795 cubic meters of gas cells filled with 66% air = 36,825 cubic meters
36,825 cubic meters of air mass = 44,963 Kg!

Maybe, you should have extended your first question.
However: But even if they did it this way when the Zeppelin descended the Hydrogen cells would be mostly empty and would fill with air causing the ship to gain weight and crash. No! The weight would be the same, as I wrote before. A balloon does not gain weight, if you inflate it with air It creates lift if you fill it with hydrogen, it looses lift if you remove some hydrogen, but if you add air, nothing happens.

@sweber But if you add water to a ship this increases the mass and it sinks.  If you add air to a hydrogen cell isn't that the same thing?

Putting water into a ship means removing air... Keep in mind: Everything displaces air, and observed weight is real weight minus weight of displaced air. If the weight is negative, it causes lift. The hydrogen displaces only half the air at groundlevel, but the air weights twice as much than at altitude, so the lift is the same. And since air weights the same as air, it contributes nothing to the weight / lift.

@sweber Maybe I don't understand a basic concept.  The gas cells start 100% filled with air.  This makes them heavy.  However before takeoff hydrogen is pumped into them which is light.  The difference gives the airship lift.  But if you did the reverse and pushed air into the hydrogen cells (as would happen during a descent with mostly empty cells) wouldn't this reduce lift until the ship crashed?

@myNewAccount: Yes, you're missing something very important.  The gas cells do not (or at any rate should not) EVER contain air, because hydrogen mixed with air is explosive.  They start out EMPTY, like a balloon before you blow it up, and are filled with hydrogen, which then expands and contracts due to outside air pressure.

@jamesqf  Thanks, now I'm starting to understand how this works.  I didn't realize hydrogen and oxygen couldn't mix.  So the rigid shell of the ship helps the gas bags retain their shape.  The hydrogen inside also helps.  The pressure of the hydrogen inside the bags goes up and down but it's always just hydrogen.  Now it makes sense to me.

@myNewAccount: Yes, though the rigid shell is more for streamlining & other structural reasons than for having the gas bags retain their shape.  You would have compartments inside the frame, each holding a bag with hydrogen in it.  The bag could expand according to the outside pressure, until it reached the size of the compartment, at which point the craft would either have to descend, or vent some hydrogen to keep the bag from bursting or breaking the frame.

The gas bags always contained close to 100% hydrogen. No air was supposed to be in them. On the ground, the bags were only partially filled because of the high air density and pressure. The rest of the internal volume of the Zeppelin was filled with air which could enter and escape the hull through the many gaps and venting openings.
Compare that to a weather ballon. On the ground, the big envelope is nearly empty and only a small bubble of lifting gas (today mostly helium, but hydrogen would work even better) sits at the top. Only that the nearly empty ballon is in clear sight, while in a rigid airship the half full gas bag is hidden behind the rigid hull covering.
One cubic meter of hydrogen weighs only 90 grams compared to one cubic meter of air at 1.225 kg under standard atmospheric conditions. With the same amount of hydrogen a balloon will have the same lift, regardless of altitude, because its lifting gas will displace a bigger volume but the same mass of air as it expands with altitude. With Zeppelins, fine tuning of lift by adding dynamic lift is possible but limited to maybe 5%-20% of the ship's mass (more for smaller and faster ships and at lower altitude). By growing larger, the mass to volume ratio of Zeppelins decreased, so the latest WW I ships could be built light enough to fly with their gas bags only one third full.
Civilian airships rarely flew higher than 2000 m high and had to circumvent higher mountain ranges because they filled their gas bags to maybe 80% on the ground. Flying higher was not considered worth the reduction in payload and fuel that would result from less lifting gas.
Compare that to the wartime Zeppelins. They were in a continuous arms race for higher altitude with the British aeroplanes which were sent out to chase them. And until 1918 the most modern Zeppelin designs  were ahead. The limited altitude of older ships also meant that many were scrapped after one or two years of service. The table below shows the maximum altitude at which the gas bags were filling out the available volume of the ships. The list below gives the maximum altitude of the best ship of each series and the density ratio shows the degree to which the gas bags could be filled on the ground to achieve that altitude. Note that they still carried tons of water ballast and ordnance when flying that high.

LZ 112 tried with speed what could not any longer be achieved with altitude: It had 7 more powerful engines where older ships had only 5 and could cruise at 131 km/h as opposed to the 100 km/h of the older ships. The diminishing returns of flying higher and the stress of spending close to a day at -40°C and without a pressure cabin or a pressure suit made higher flight altitude impossible. What also held progress back was the size of airship hangars: Since LZ 62 the maximum diameter could not grow beyond the 23.9 m of the last wartime ships, so their volumetric efficiency went down when they grew larger by adding length.

Peter thank you so much for your information.  After thinking about your answers I think I understand.  The hydrogen cells always displaced a more or less fixed volume of outside air.  Airships were rigid and because the bags were attached to the frame they retained their shape and displaced approximately the same amount of outside air even when they were half filled with hydrogen.  Is that more or less correct?  Although a pressurized cabin would have been heavy I wonder if it would have been worth experimenting with since Zeppelins could reach 7,000 or 8,000 meters.

@myNewAccount: I hope you meant to say "The hydrogen cells always displaced a more or less fixed mass of outside air". The outer shape was rigid and the gas bags had variable volume, depending on atmospheric pressure. One big advantage of Zeppelins was that you could walk around the ship. Each engine had mechanics tending them during flight. Repairs of the engines and the structure were done in the air. Pressurizing all those stations was prohibitively expensive in terms of weight.

maybe I didn't understand.  I meant volume.  For example the rigid hull and attachments force the gas cells to stay + or - the same approximate shape regardless of altitude and pressure.  Lets say the gas cell displaces 100 cubic meters of air.  So at sea level it displaces 122 Kg of air.  At 5,000 meters it displaces 68 Kg of air.  This gives a rigid airship it's maximum altitude.  I hope I understood that right.

@myNewAccount Unfortunately, no, this is not the way it works. The gas bag is floppy, just as the partially filled balloon envelope is. Lifting gas forces it upwards against the structure and straps keep it connected to the sides of the frames, but at the bottom it is empty and front and rear are pressed together by atmospheric pressure. When the airship climbs the lifting gas expands and fills more of the gas bag until the gas bag is completely full at maximum altitude. Climb higher and the emergency valve opens, releasing excess hydrogen. That way the gas bag produces the same lift,

@myNewAccount … regardless of altitude. It will displace more air volume as altitude groes up, but always the same mass of air. Outside pressure dictates what pressure and density the lifting gas has because the gas bag is floppy. At least when altitude change is slow enough so temperatures inside and outside the gas bag are equal.

Maybe I don't understand the specs.  The U Class Zeppelin's cells displace 55,795 cubic meters of atmosphere and create 64,750 Kg of useful lift.  At what altitude is the 64,750 Kg of useful lift accurate?  At 1.22 Kg per cubic meter that looks like sea level.  But would that be true at all altitudes?

@myNewAccount: The hull displaces those 55795 m³, but the bags inside are flexible and the volume they took up depended on how much hydrogen was in them and the atmospheric pressure. They would displace their maximum volume at maximum altitude and proportionally less at lower altitude. A ship with full gas bags on the ground would not be able to climb, or it would lose lift when venting lifting gas. A Zeppelin was never filled up to create those 64750 kg of lift, they were just filled until the ship was balanced. For an U-class this would mean the bags were half full on the ground.

Fascinating, I think I'm starting to understand.  Generally speaking if the fully loaded Zeppelin weighed 25,000 Kg would the ground crew fill the hydrogen cells until they produced roughly 25,000 Kg of lift?  In that way the ship would be balanced and most climbing and diving would be accomplished with the elevator?

@myNewAccount Please don't hesitate to ask a new question - comments are poor for explanations. Yes, only as much hydrogen as needed to lift the loaded ship would be allowed, or the ship would have risen uncontrollably up to its maximum altitude (Prallhöhe in Zeppelin speak). The elevator was only for controlling pitch, steering up or down was done with the engines. At low speed, the elevator function is even reversed.

@myNewAccount:  When the lifting gas is at a temperature equilibrium with outside air, the same mass of hydrogen in the bags will keep the ship balanced at any altitude. With altitude the volume of this mass expands until the gas bags fill all of the ship's available volume at maximum altitude.

Some physic experiments
Let's assume we have a 1m³ container of negligible weight at sealevel. It displaces 1m³ of air, and this air weights 1.22kg.
When filled with hydrogen at ambient pressure, this hydrogen weights 0.08kg. The lift is 1.22kg - 0.08kg = 1.14kg.
Now let's bring the container to an altitude where pressure is just 50%, and let's ignore temperature. 1m³ air at that pressure weights 0.61kg. With the hydrogen, still at full pressure, the lift is 0.61kg - 0.08kg = 0.53kg.
Let's now bleed half of the hydrogen, until it reaches ambient pressure. The remaining hydrogen weights 0.04kg, and the lift is 0.61kg - 0.04kg = 0.57kg. It increased by just 0.04kg!
The container is now brought back to ground. The lift is now 1.22kg - 0.04kg = 1.18kg.
Next, 0.5m³ air is let in, to equalize air pressure, and the hydrogen compresses to 0.5m³. Overall, the container now contains 0.61kg air and 0.04kg hydrogen, the lift is 1.22kg - (0.65kg + 0.04kg) = 0.57kg. This is the same as at altitude!

What does this mean for airships?
If the entire volume of an airship is filled with hydrogen at the ground, it would loose about half the lift at that altitude. Since it can't withstand the over-pressure, it had to bleed off half of the hydrogen, which doesn't change lift much. And if it returns to ground, letting air in to maintain ambient pressure, it would still have just half the initial lift, not more, not less.
Instead, the volume of an airship is filled partially at ground, and the gas is allowed to expand at altitude. This way, it starts with low lift, but it maintains this lift during the entire flight.
Rigid airships like those in question contain gas cells inside the hull, which were inflated partially and could expand during flight.
For non-rigid airships (blimbs), it's usually quite the opposite: The lifting gas is directly under the hull, and there are cells filled with air.
Now, looking at your numbers: The LZ95 weights 23.000kg. This needs about 30% hydrogen to fly. Add some more hydrogen, and it can carry some load.
I guess

Maximum lift is when filled 100% with hydrogen, though it can't climb to higher altitude
Maximum altitude can be reached only with almost no load.

Remember: An aircraft has a maximum speed, maximum load and maximum range. But not all at once.

Your last line is absolutely correct. When I read the specs on ww1 - ww2 aircraft if they carry a lot of bombs they probably aren't flying very far.


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Can you weld a propeller blade from 120 degrees to 180?
My uncle flew for Mission Aviation Fellowship (maf.org) for a university in Texas, and I believe that he was flying a Cessna to the Amazon (among other places). He actually knew Jim Elliot (who went to the Waodani of Ecuador), the martyr about which the movies "Beyond the Gates of Splendor" and "End of the Spear" were made. He had landed in the Amazon and one of the vanes of his 3-vane propeller broke and flew somewhere off into the jungle. So he got the welder out of the back of the plane, and cut off one of the remaining two vanes and welded it back on at 180 degrees to the one remaining vane. Then he took off and flew home. I love this story, but I always wondered...
How often does a propeller vane just fly off like that?
(And how challenging was that repair?)
Finally, was there any maintenance that would have prevented propeller breakage?

EDIT 1:
From a page from his own memoir that I found online (search for Strash), I found that he definitely flew a Cessna 206 while in the Congo near Kinshasa (then Zaire) in Africa.
Before he worked in Africa, he worked in Suriname in South America, which, indeed, is considered to be part of the immense Amazon jungle. I don't yet know the specific plane he flew while he was there, but a direct contact at the MAF tells me it may have been a Cessna-185 or a Cessna-206.
My mom seemed to recall the story clearly (even though she's 83 and bedridden and I'm caring for her 24/7). Mom said that there was a tree branch that was hit during takeoff, which took off the blade.
If I can get a copy of Uncle John's memoir, the story may be in there.
Bottom line, I know the story was actually told because I received it personally while in his presence and he would have set the record straight at that time if it didn't happen. My recollection of the details may be fuzzy (meaning Amazon or Congo) because that was perhaps approximately 1990 when he visited us in New Jersey.
When this story took place, he may have been in either of those two places, Suriname or Congo.
Here is an early picture of my uncle from his yearbook online, before he became a full professor. He worked both at the MAF, and as a professor teaching Aviation at LeTourneau University, Aviation Technology division.

EDIT 2:
I have to apologize... My story was probably wrong all along. Contacting someone related to MAF directly yielded what I consider to be a more realistic story, which I include here because it's interesting in its own right:

I was unable to verify your uncle’s propeller accident, but when
asking around, one person remembered a similar story but it involved
John flying a Cessna 185 in Africa where he lost a propeller tip (no
welding involved). In order to fly the plane to an MAF repair base, he
reportedly sawed the other propeller blade down to match the shorter
one and it flew reasonably well out of the jungle until it could get
properly fixed—no third blade involved. Of course, this cannot be
confirmed, nor did it comply with MAF standards of safety, and it was
60 + years ago.

Now all of your responses make more sense to me. Thanks, everyone!

Sorry but this story seems extremely implausible. If you lose an entire propeller vane... er... blade, the engine will leave the airplane with the imbalance.  What is survivable is losing an inch or so of propeller tip where you shut the engine down right away and glide to a landing.  In that case you would just cut off the tips of the other blade or blades the same amount and fly it like that.

@JohnK -- I did get the story from him personally, but I think that he has since passed away, so I can't confirm the details directly with him. But another relative might. I am just passing the story along as it was told to me. I do believe in God and angels, so it would not surprise me if there were something special about exactly how it happened, like the blade falling off completely during a particularly jarring landing, and then the blade flying off taking a lot of the energy. I do believe there was a lot of vibration after the blade was lost.

Usually, once...

Well, the story as you detailed it is impossible, knowing the way propellers are constructed.  A entire blade coming off even on landing would rip the engine from its mounts. And 3 blade propellers are complex machines with blades that are fixed in a hub with bearings so they can rotate. Such a welding repair is not possible. He was embellishing somewhat to enhance the story. It true at all, was likely a inch or so of tip breaking off from metal fatigue, which can happen. If that happened I'd hacksaw off the tips of the other 2 blades to get close to balance again and fly it out.

@JohnK -- I can't prove it, but you can't disprove it either. Let's leave it at that. I want to believe my uncle's story.

At what stage of flight did the propeller blade depart?

It seems the answer you want is "as often as God and the angels want them to." But for that christianity.stackexchange.com might be a better fit.

I can't prove it, but you can't disprove it either. That's a bit of a weebly-woobly standard to have. Anyone could come at you with weirdest tall stories and you'd most likely have no way to disprove them. Inability to disprove something has just about nothing to do with how credible something is. It's in fact a common fallacy.

@MicroservicesOnDDD it's perhaps entirely possible that some small piece of one blade broke off (perhaps a tip), and then, an incredible "bush fix" was made one way or another. (Which would be a great story.) These stories tend to grow and change through time from the beginning.  For example, we have a family story (WW1 type era) involving an amazing military act, and one part of the story is "... and he got the Victoria Cross for that".  However, it turns out it wasn't literally a VC but just some other "hellova good medal!"  It is common and normal that stories get distorted like that..

..one well-understood issue in "story transmission" is lack of arcane expertise of the tellers. For example, "there was a problem with the yaml in the ci/cd" becomes "the code had a bug!" or some subtle medical issue becomes "he got cancer!"  For example it could be that some other component (perhaps a critical bolt or something, I know nothing about props) flew off in to the jungle, not "a blade".  Someone  retelling the tale who knew nothing about aircraft engineering said "a part" flew off, soon enough "a blade" (what other word would come to mind to a non-expert?)  So, just a thought!

It is entirely possible that God could miraculously allow such a repair; I have no doubt that He is able to do so, or that he would do so. However, because this isn't a site about Christianity, answers are going to focus on what is possible assuming no miracle takes place. Miracles are, by definition, impossible or at least extremely unlikely barring divine intervention, so if this was indeed a miracle, it would be expected that experts would find it highly improbable.

Hi everyone. Please see EDIT 2 in my updated question to see that it was only the tip of one blade that was lost.

This is a fine story, but:
When you see real propeller hubs, you'll quickly understand that one does not simply transform a 3 blade into a 2 blade with simple cut and weld methods:
 (picture source: e-bay,
Smokin Rivet Airparts & Salvage LLC)
As you can see, the hub is not solid. cutting one seat off would leave enormous gaps to fill, and  while with proper equipment and environment that would be possible, in field it is not. Attaching the blade directly into the "valley" of the hub would be just as problematic.
As pointed out in Sophit's answer the weldability of the hub material and availability of proper equipment is questionable to say the least, and aligning the blade properly will be a chore. Then there's the imbalance caused by the lost blade as per Chris' answer, which will render the plane pretty much unflyable, quite likely by ripping the whole engine off.
As you stated in your comments, you can't prove this story, but it's pretty much debunked here.

While I'm sure what you say is true for what you have shown, this is most likely a more modern development, and when they started MAF shortly after the war, you would have to go on the attachment technology of the day, and I'm sure there's more than one way to make a propeller, or attach blades to it. You would have to show me every way of attachment since World War II to convince me. And somewhere in the showing, my bet is that you'll find a weldable one used by the bush pilots, probably because of its ruggedness, and perhaps for its fixability.

Not gonna argue with you on this. If welding was involved, this is the basic hub design in question, they have not changed since forties. You updated the question to include information that the blade was "lost" during takeoff, there simply is no way that the plane would remain flyable. Full power setting combined with the imbalance (weight & thrust) would rip the engine right off the plane. Is it possible the blade was lost during taxi? Then again, the modifications depicted in the story still aren't plausible.

https://www.aviationart.com/products/1940s-14ft-goodyear-super-corsair-f2g-1-3-bladed-hamilton-standard-6491a-14-complete-airplane-propeller

Thanks for letting me see for myself. I hear what you're saying, and now I can see what you are saying, but I'm waiting for my full investigation in seeking out the memoir of my uncle and various books and family member information sources to decide.

The full investigation is in! See EDIT 2 at the end of my question. The tip of a blade was knocked off instead of the entire blade. My uncle then had to even out the blades in order to take off again.

A blade of a propeller breaking off in flight is rare, but not unheard of. It can be catastrophic- the imbalance can tear the engine straight off the airplane, and this can happen very quickly.
Unless the blade separating was due to some acute trauma (i.e. hitting something), it's almost certainly a maintenance issue. Either some unnoticed damage to the propeller was allowed to worsen, or the blade was improperly attached to the hub in the first place.
The repair sounds extremely implausible:

Propellers take the most abuse of any component on the airplane- centrifugal forces on the base of the blades are in the tens of tons. It seems likely that a field weld would be much weaker than the original propeller and could easily just cause it to separate again in flight.
Trying to fly with 2 blades instead of 3 would result in dramatically lower available power (without overspeeding the engine, anyway). Light aircraft don't have that much power margin- I find it unlikely that the plane would be flyable even if the weld held.
While having the blades at 180 degrees from each other would reduce the imbalance, it would still be large if the blade were not attached perfectly.

I can't prove it, but you can't disprove it either. Let's leave it at that. I want to believe my uncle's story.

@MicroservicesOnDDD - https://en.wikipedia.org/wiki/Russell%27s_teapot

@MicroservicesOnDDD You're welcome to believe anything you want. I don't see how this repair would be possible, but like you say I can't definitively say it's impossible.

This story is not plausible.
The location would have been Shell Mera in Ecuador.  The Amazon is large, but Jim Elliot worked in Ecuador.  Several different planes have been used out there.  The Piper that was used during the fatal flight was a yellow Piper 14 Family Cruiser, which was light.  Several other planes have been used since in Ecuador.  Amongst others a Cessna 180 and Cessna 206 Turbo.  These are relatively powerful piston engined planes.  The propeller would spin at about 3000 RPM, but the torque during takeoff would be enormous.
Reasons why this is not plausible:

The propeller blade that departed the plane would probably wreck the plane by breaking the motor mounts.
The fracture would not be a clean one, making it extremely difficult to fit another blade to the fracture.  As a matter of fact, the fracture would in itself cause an imbalance, unless it is cleanly cut off using a lathe.  The Cessna 180 has a constant speed prop.  The governor would also not work.
The blade is a profiled aluminum blade.  A weld would only cover the surface, not the core.
A welder in the cargo of the blade is very unlikely.  These are extremely heavy loads, as it also needs to be fuel powered, and unlikely a electric transformer type.  You don't just tag one along like a wrench, just in case you need it.  If it would need electricity, you would be out of luck after an emergency landing, and if it would burn fuel, it will be a fire hazard.  If it was in the cargo load, then it's unlikely that there would be aluminum rods to weld, as everything out there would probably be steel. Aluminum welds are hard, and probably need MIG/TIG including the needed gases.
(Near) crashes were not uncommon.  However, there were always alternatives to get the planes out, for instance with broken gear legs, flat tires, even cylinder failures. The procedure would be easy: call to the base station, describe the needed parts and fly in assistance.  Then repair the damage in an authorized way, and fly it out after testing it properly.

I spoke to a retired MAF pilot that was stationed at Shell Mera in Ecuador, and he could probably confirm that this would not have happened.  He also wrote a book about his work in Ecuador, and while he told me some wild stories and wrote it all up, it would be hard to explain why he left this story out. There can only be one reason in my mind.  It didn't happen.
Can you share more details like the approx date, location and type of plane or even registration?

Just checked with my source. He is not aware of it.  However, he does recall hearing about an incident in which a MAF pilot in Zaire / Congo, involving a two bladed prop, of which a tip broke off.  He sawed of an equal part of the other blade, and flew home. with a heavily vibrating engine.  At arrival all crankcase bearings were shot.

I think it's safe to say that this repair would be impossible with a constant speed prop. Do you happen to know if there were many planes equipped with fixed pitch props in the area at the time?

@Chris-RegenerateResponse  Nothing stopping you from doing that with a CS prop. A blade tip is a blade tip.  Flying home with missing propeller blade tips after sawing off the opposite side has been done in the Canadian bush, and I'm sure all sorts of places over the years. If you could remove the prop and had a way to suspend or support it on its center axis, you could achieve a fairly good static balance with lots of filing. Otherwise It comes down to making the finished cut as close to the other side's length as possible.

@JohnK Sawing off a blade tip is fine. Removing one blade and putting it somewhere where there never was a blade to begin with, like in OP's story, is obviously impossible with a CS prop. If you just weld it to the hub, its pitch obviously wouldn't change, and I can't imagine a plane with one variable pitch blade and one constant pitch blade would be flyable.

I thought you were just talking about sawed off tips. Of course. See my comments in the OP.

One correction (possibly): there are very compact and light alternator driven stick welders, although I'm not sure that was the case back in the days. One could weld aluminum with such a device, but as stated many a times here, not a propeller hub. Just nope.

@pnauta -- Sorry about the confusion, but your assumptions are wrong. He did know the Elliots, and may have made several flights there (to Ecuador). But he worked for a mission organization that flew to many places. I'm trying to contact family now to verify as much as I can, but I do believe the blade flew off as he landed some place in the Amazon jungle. But I'm working on verification...

Typos: "A welder in the cargo of the blade ..."; "If it was in the cargo load, ..."

Propeller blades are made either of high-strength aluminium alloy, wood or carbon fibre. None of those materials can be welded, except aluminium but only with very specific equipment (using argon as shielding gas and welding rod based on aluminium).
Just remember the story as it was told and don't ask yourself anything else

I could see propellers being made of hollowed steel with a balsa-wood core, especially when facing flocks of geese or other birds. Are you sure steel has never been used? Landing frequently in the jungle, I would want a particularly tough and resilient propeller.

@MicroservicesOnDDD: the blades can have a so called abrasion or wear strip on the leading edge made of some steel alloy but that's just for protection against dirt. Steel would be susceptible to corrosion. I don't think it has ever been used (except maybe at the beginning of the aviation history when aluminium was too expensive). Anyway the better material for propeller is wood and carbon (or glass) fiber.


text:
Will driving experience benefit flight training?
I think my question/situation is a little bit of rare so I start this question and want to hear from you experience pilots.
I'm a 30+ migrant from China working as a software engineer for 10 years. Now I use my weekends taking flight training for Private Pilot Licence in Sydney. Currently I don't see any big problem with my training and probably in next couple of weeks I will have my first Solo. What concerns me is my past experience. I can say I'm rather a good programmer and writing code for computer(iPhone) is really different from driving or flying. My flight instructor tend to say (and I agree) I always fly with the exact numbers rather than make necessary adjustment according to situation outside. So I guess with the past 10 years indoor in-front-of-keyboard office life/experience. The more you get your life/experience aligned with computer/coding, the harder you could get out of it.
What's worse is I don't have any driving experience until two months ago I started to practice for my Australian driver's licence. Now I have my licence issued for one month and I still only drove four or five times with rented car. Though I love public transportation and I'm not that declined to own a car, now I start to thinking about the benefit of driving experience for my flight training. At least driving skills are accumulated with "hours" (which is the same as flying) and it will get me out of computer world.
Do you have any suggestion or advice on this? Thanks for reading and replying. Cheers~
P.S. if possible, my target is CPL one day.

Welcome to the site, it's good to hear of people taking up flying. Your question is not completely clear. Are you concerned you lack decision making capabilities, or are you thinking that somehow driving experience translates to flying experience?

Thanks @GdD. I just think(guess) driving a car and an airplane are similar. My previous experience limited my understanding of developing this kind of skill. For a not so accurate example: in my previous working environment everything is "known", the requirement the language spec the algorithm. But with driving/flying, one has to observe somewhat "unknown" or "unpredictable" situation like passengers on road or "flying too low on base". it's just not "a given input will must result a definite output". I think i've lost somewhere there.

I don't think driver training is beneficial to flight training, the physical acts of driving and flying are too dissimilar. Driving is a handy skill to have for sure, it's usually much easier to get to the airfield for one thing, it's not going to translate to flying though.
As for flying the numbers you are falling back on what you are good at: attention to technical detail. Attention to technical detail is a very good thing in flying - having a plan and rehearsing for emergencies can save you neck. What your instructor may be telling you is that you need to focus your attention to other details. You may be spending too much time looking at your instruments and not enough looking outside. I did this at first because I had spent a lot of time playing flight simulators and I'm also in IT and spend a lot of time looking at displays in front of my face. The cure is simple: force yourself to get in the good habits of looking outside the aircraft and developing situational awareness. You can only get that through flying practice.
It sounds like you are concerned that you don't have a sort of instinctual "feel" for flying. Everyone has a different set of aptitudes, some people are better at certain aspects of flying than others. If there's an area you are weak in then focus your training and practice on developing it into a strength.

From my subjective experience, training pilots without driving experience is not a big deal, as long as it not an indicator of being uninterested in things technical. (Remember, tons of people start flying gliders in their teens, before ever having driven a car).
What you are describing is the typical path of the flight student, and from the fact that you are concerned with making good progress: a motivated flight student.
Without training and experience, you can not expect yourself to intuitively know what to do when you are "too low on base". In fact, you can not even expect yourself to even know that you are "too low on base". The only way to compensate for this, in the beginning of your training is by doing things "by the numbers" (ie, knowing what height you should be over the blue house etc) (sort of like following an algorithm).
That being said, the most common (once again my subjective perspective) hurdle seen in new pilots with plenty of flight simulator experience is underestimating the importance of outside visual references (in practical terms: they don't trust their own ability to judge their attitude looking at the outside horizon.
For an educating experiment: Ask your flight instructor to oscillate the aircraft in pitch +/- 1 degrees while you look only at the artificial horizon, then have the instructor repeat it, while you look outside, at the natural horizon, and make your own conclusions as to which is "the better" instrument. :)

My experience is the other way around: Learning to fly improved my driving skills. This mainly in the field of trip planning and discipline of execution. Remember, with a car you can always pull to the right (in Australia to the left) and stop. With an airplane you do not have this luxury, so you learn to think ahead.
If you are comfortable with public transport, forget to get a car. Driving is expensive. But try to get behind the wheel every now and then to build up experience.

Driving experience helps in flying only by "curing" your reflexes. Nothing else. You can have no driving experience and still be a good pilot, but there is a thing called "Pilot Aptitude". Some people are good at it, some people are not. This does not come from how experienced a driver you are, but comes intrinsically from the fact "How good you are". Many air-forces across the world have this test, and it is often claimed that one cannot gain this aptitude just by flying more. You need to have this aptitude, and after that you can improve your flying skills by having more experience and practice.
Indian Air Force conduccts PABT

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/


text:
What are these hooks on the A320's wing used for?
I was flying on an A320 (equipped with sharklets) the other day, and I found these hooks on the wing's surface:

Note that the single hooked attachment is outside the wing's "DO NOT WALK OUTSIDE THIS AREA" line. What are these hooks used for? Why is the single hooked attachment outside the "safe" walking area line?

my first guess would be attachment points for the emergency slide

https://youtu.be/YEXTJDhGOSQ
Hudson landing talk about them

At 2:00:30 in video above. Comment from passenger

In the event of a water evacuation, there is a line associated with each overwing exit that is extended, and clips to that yellow bracket, as a guide to get everyone out on the wing.
From Wikipedia, these are for overwing exits.

The use of overwing exits in a ditching varies from airline to
  airline. On aircraft fitted with overwing exits, there is typically a
  raised escape rope bracket (about a third of the way from the door)
  attached to the wing's upper surface and typically painted yellow.
  This is accompanied by an escape rope found in the frame of the exit
  after opening the hatch. Once this rope is attached to the escape rope
  bracket, it will aid passenger evacuation onto the wing to await
  rescue or to enter the water (depending on the airline's procedures).
  On aircraft with life rafts to be launched via the overwing exit, the
  wing will be used to provide a boarding platform onto the life raft.
  On certain regional aircraft, the overwing exits are the only escape
  route in the event of a ditching.

Thanks, this is what I was looking for. Isn't having ONLY overwing exits on regional craft a bad idea if a wing mounted engine catches fire or if the turboprop hasn't stopped spinning?

Most larger airplanes have several doors, not just over the wings. Depending on the situation, some or all can be used. In case of Flight 1549, the overwing exits were the most important ones.

@shortstheory "... the only escape route in the event of a ditching" (emphasis mine). If you've ditched the plane, the wing-mounted engines are underwater, which will extinguish the fire.

I can't believe I never doubted my assumption that these were lift points for removing the wing for maintenance.

@ToddWilcox Do you know what: I thought they are yellow because the Airbus I first noticed them on was a Lufthansa one and that's (one of) their CI colours.

@DavidRicherby: So do the forward entry doors magically jam shut in a ditching?

@Sean If you disagree with the quote, take it up with its author, not the person who quoted the person who quoted it.

Likely they are tethering points for emergency exit ropes. The Boeing 737-800 has the same type of hooks. See top row from the water evacuation procedures:

(Image source)

From this analysis of Hudson ditching, in which life lines were not extended by passengers opening the overwing exits: "Despite a regulation requiring the life lines at overwing exits [...] circumstances in which they could be used effectively after ditching have been unclear [...] The NTSB concluded that life lines could have been used to assist Flight 1549 passengers on both wings, “possibly preventing them from falling into the water.”"

The inner ones are mounting points for evacuation slides and/or a rope handrail since the wing is likely to be slippery and wet. Several aircraft including the A320 and B737 have additional overwing life rafts when ditching. You clip in a rope either at these points or in a hook at the window, throw it overboard, and give it a sharp tug to inflate. Joyous passengers will then board the life raft.

Source
The outer one works as a harness securing points for the maintenance guys working on the wing, but I guess this could be used for the first purpose as well if you would want to gather everybody around one wing or something.

Perhaps the OP should read the safety information card in between looking out the window and taking pictures of the wing. ;-)

Bordering on mean.  ;-)

@SportsRacer: Many passengers who ditched on the Hudson exited the aircraft without life jacket (location on label overhead) or flotation seat cushion (as prescribed in safety card). "17 % watched most of the preflight safety demonstration, 13 % watched some of the demonstration and 8 % read the safety information card" (from analysis). The fact nobody died during the evacuation is actually a second miracle due to many "fortuitous circumstances", according to the NTSB report.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/


text:
Can an instrument procedure be intercepted from the middle and how?
While I was reading FAA-H-8083-6 I was wondering if a pilot can be instructed to intercept a procedure from a point somewhere in the middle of it.
Specifically:
A pilot departs from an airport having assigned the FOBAR1 SID. There is RADAR service, hence after departure they are vectored to a position outside of that SID. Can the ATC instruct for any reason the pilot to resume the SID? And if this can be done, how is it done? Simply Resume FOBAR1 SID or more specifically eg Fly heading 125, resume FOBAR1 SID or Proceed direct ABC, resume FOBAR1 SID?
NOTE While I mention an FAA document I am mostly interested for EASA and/or ICAO regulations/procedures.


text:
How are $V_r$ and $V_2$ determined?
$V_r$ is the speed at pilot should initiate rotation on take-off and $V_2$ is the safe climb-out speed. Pilots determine them for each take-off by looking them up in a table in POH or QRH or somewhere.
But somebody had to fill those tables in using flight tests and calculations. What are the criteria by which the values are selected?

The limits of the $V_{r}$ and $V_{2}$ are determined by the regulations in place. The rotation speed is the speed at which the pilot can the aircraft rotation for take off even when one engine is inoperative.
Consequently, this speed should be more than the minimum control speed with one engine inoperative, $V_{MC}$. The current regulations stipulate that the $V_{r}$ should be at least 1.05 times $V_{MC}$.
The safe climb out speed or takeoff climb speed is the speed which the aircraft should have reached at a given altitude. As per FAA regulations, this height is 35 feet above the takeoff surface.
The climb out speed $V_{2}$ should be at least 20% greater than the stall speed $V_{s}$ and 10% greater than $V_{MC}$.
The FAA regulations prohibit lifting the nose landing gear before reaching $V_{r}$ and retracting it till the aircraft is completely airborne.
All aircrafts undergo these tests during their certification process and these data is used by the pilots. Another thing is that both $V_{r}$ and $V_{2}$ are given in terms of calibrated air speed.

IIRC the other factor is the current atmospheric pressure as this has an impact on lift, hence they are not constant for a given aircraft or even for a given airfield and the pilot has to consult a table or computer.

@SteveBarnes, no, effect of atmospheric conditions on lift is covered by use of indicated (or calibrated) speed. There is still effect of weight and effect of atmospheric conditions on engine power.


text:
Is it possible to handle naming duplictates for teminal points in ARINC-424?
In ARINC-424 17 specification, under 7.2.6 Terminal Waypoints subsection A the "Airport/Heliport or Runway/Helipad related Terminal Waypoints" naming conventions are described. It describes the steps to be taken in order to avoid duplicates.
For example: a final approach fix (FAF in short) on runway 36 for example, will be named FF36 which works either for "Single Approach Procedure" or approaches that share a "common waypoint". If the common waypoint idea can't be applied, then the second letter becomes the route type described in section 5.7. So the FAF for a VOR approach on 36 will be FV36, for an ILS on 36 FI36 and so on and so forth.
But here lies the problem (which unfortunately I am currently facing): what if there are more than one approaches of the same type on the same runway that can't share the same points? I have 2 VOR approaches where all points IAF, IF, FAF etc. do not coincide. According to the spec:

waypoint identifiers will be unique only for the airport specified; a waypoint identifier used in a terminal area cannot be repeated in that terminal area

I could get away with it as there are is no parallel runway and I can start adding numbers (eg FV361, FV362 etc.) and it won't make any significant difference as these points will be used for simulation purposes and I have a couple of ways to "cheat". Though knowing what happens in reality will add to the simulation realism.

Which airport is this?

@Bianfable it's LGEL but unfortunately I can't provide you with maps as the one of the 2 approaches in question is in military AIP and is not public. The other can be found here (link will most probably expire in the next AIRAC update). Let me know if you find the one published helpful and I will update the question.

OK, my navigation data for LGEL shows 3 VOR approaches: VOR Y RWY 36L, VOR A RWY 36L and VOR W RWY 36R. Not sure if any of these is the military procedure you mean. I could write an answer containing the names of the CD and FD waypoints on these procedures. Would that help?

Duh I forgot. They are doing works and they converted a taxiway to runway and named it 36L. Sure the names for  VOR Y RWY 36L and VOR A RWY 36L will be helpful as long as they don't have coinciding points. I only have VOR Y not VOR A so I can't tell immediately if it will help.

Based on your example of LGEL, I found 3 VOR approaches in my navigation database with the following points:

Approach
Cx Name
Fx Name
Mx Name

D36LY
CD3LY
FD3LY
MD3LY

D36RW
CD36
FD36
MD36

VORA
CD36
FD36
MD36

As you can see, the VOR W RWY 36 R approach (where you also linked the chart) uses the normal naming convention for a runway 36. The VOR A is the circle to land approach using the same waypoints. The interesting one is the VOR Y RWY 36 L approach, which uses different waypoints. The naming now uses only the first digit of the runway number (3 instead of 36), followed by L for 36L and then the letter of the approach (here Y).
This allowed me to search my navigation database for similarly named waypoints and sure enough I found a good example, where there are 2 approaches to the same runway using a different waypoint at LIRP (Pisa San Giusto):

Approach
Cx Name
Fx Name
Mx Name

D04LZ
CD0LZ
FD04L
MD04L

S04LY
CD0LY
FD04L
MD04L

As you can see, both approaches use the same final approach fix (Fx) and missed approach fix (Mx), but a different final approach course fix (Cx) using the naming convention described above.
The same convention can also be applied when there is no left/right/center. E.g. EBBR (Brussels) uses Cx waypoints CD25Y and CD25Z for runway 25.
My copy of the ARINC 424 standard does not describe this convention, but apparently this is what Jeppesen uses.

Ah indeed this is ingenious. +1 from me. I hope you don't mind if I wait for 1 day before accepting. I'd like to avoid discouraging someone else from answering, if they have another bit of information to add -though I doubt it :).

@SteliosAdamantidis Sure, no problem. Maybe someone will find where this is actually documented...


text:
What are the effects of the Boeing 787's very flexible wings?
I recently came across this picture of the Boeing 787 series aircraft's incredible wingflex:

I suppose this is a consequence of using very light CFRP wings, but how does the wingflex itself improve the 787's flight performance? Do the benefits/drawbacks also apply to the 747-8 (which IIRC also uses CFRP wings)?

Not an answer, just a nice video on really incredible flex for the DG-1000: http://www.dg-flugzeugbau.de/Data/Videos/bruchversuch-i.wmv. They also do that for the big ´uns like the A380, which is really fearsome (but I don´t have a video link at hand).

Related: http://www.airliners.net/aviation-forums/tech_ops/read.main/253605/1/

@yankeekilo thanks for sharing, that was a pretty cool video. I heard that they stress the CFRP wings a lot, but not to breaking point as the shrapnel from a CFRP wing could be quite severe.

Just found: http://www.airliners.net/aviation-forums/tech_ops/read.main/267122/

Actually, the do break them: https://www.youtube.com/watch?v=sA9Kato1CxA or https://www.youtube.com/watch?v=z19m9LZOOZY. Those tests are huge and of course lots of safety precautions have to be taken.

That's a dreadful image. I really doubt the wings increase in length dramatically as they flex. The wingtip's motion would surely describe something closer to an arc than a vertical line.

@RedGrittyBrick excellent point; I felt something was amiss, but I couldn't figure it out. Thanks!

From here:

The amount of flex is really a product of the material. The wing requires a specified ultimate strength; with metal, that translates into a given amount of flex. This can be varied within limits, but it is really the material, its stiffness to yield point ratio, and its fatigue properties, that control how much flex you are going to end up with. CFRP is a very different material, and has much less stiffness for the same yield point, and has essentially no fatigue problems. This is beneficial in that it provides a smoother ride in turbulence; the wing acting essentially like a giant leaf spring. There is some lift lost due to the nature of the curvature, though. However, this is relatively small.

How do you correlate yield and stiffness? CFRP has a higher specific stiffness compared to aluminum and steel...

Increasing stiffness, means increasing mass, means decreasing yield/lift.
This material provides high strength with a relatively low stiffness/mass, meaning a good ratio and, in turn, the consequential flex that you see.

But the flex is in the design, not the stiffness of the material. You could build much stiffer wings with CFRP. CFRP (done properly) offers both excellent stiffness&strength, with relative low breaking strain compared to aluminum. I agree on the fatigue point, though.

You could build much stiffer wings with CFRP. However, the increase in mass will reduce the resultant lift more than having 'flat' wings.

My point is that CFRP does not in general show a lower stiffness for a given yield. The flex is a design decision giving the best compromise, but not inherently due to the material.

The wings of the Boeing 787 are so flexible because its carbon fiber material can be stretched more, and the high aspect ratio of 11 will magnify this effect. In flight, all you will feel is less shaking due to gusts, because the wing will dampen load changes more effectively. On the ground, the wing might have less tip clearance, because less in-built dihedral is needed - the rest is supplied by the wing's elasticity in flight.
The influence on performance is slightly negative, but this is a very weak effect. It can be compared to the rolling resistance of a stiff bicycle versus one with a spring-loaded frame.
The amount of bending for a given bending moment depends on three factors:

Wing span: A given curvature of the wing due to bending at the wing root will cause a tip displacement which is proportional to the distance of that tip from the root.
Spar heigt: This curvature grows with the inverse of the square of the spar height. A lower relative thickness of the wing will produce more bending.
Spar material: The Young's modulus of the material describes how much it stretches for a given stress. More important, however, is the elastic elongation at yield stress. Carbon fiber has a higher Young's modulus than aluminium, but is elastic until rupture, so it can be stretched more and produces more bending at yield stress.

The numbers: The Young's modulus of aluminium is fairly constant for a wide range of alloys and normally 70,000 MPa or N/mm². The modulus of graphite fibers depends on their manufacturing process and varies between 200,000 and 700,000 MPa or N/mm². However, this value cannot be compared directly to that of aluminium. The final modulus of the composite depends on fiber orientation and resin content.
It is safe to assume that Boeing (or more precisely, Mitsubishi Heavy Industries) uses a modern, high-strength fiber like IM7 (pdf) (IM stands for intermediate modulus), which has a modulus of 276,000 MPa. It is also safe to assume that most of the fibers are oriented in span direction, so they can contribute fully to taking the bending loads. If we assume a conservative fiber content of 60%, the resulting modulus of the spar material should be 164,000 MPa. However, the spar is not a discrete component, but part of the wing box which also has to take torsion loads. While Aluminium is an isotropic material (it has the same properties in all directions), CFRP is highly anisotropic, and adding torsional strength will require additional fibers in other directions. Consequence: The effective modulus of the wing box in bending direction might be as low as 110,000 MPa.
In the end, what counts is how much material is there to carry the bending loads. Here the yield stress of the material comes into play: The more stress a material can tolerate before it shows plastic deformation, the less of it is needed to carry a given bending moment. To arrive directly at the maximum deformation, it is enough to look at the maximum elastic strain. With IM7, this is 1.9%, and with high-strength 7068 aluminium (pdf), it is less than 1% before the material suffers permanent elongation. This means that, even though CFRP is stiffer than aluminium, it can be loaded more and will stretch more before it reaches its limits.

Thank you for the answer. But my question was about the in-flight performance of the extremely flexible wings, not about why the wings flex in the first place.

@shortstheory: Theoretically, there is a small performance reduction due to wing flex, but this is extremely small. My point is that it mainly reduces the load factor felt by the payload due to gusts.

But Airbus A350 that is developed using almost same materials, has the same wing flex or not? and, if not, simply "why"?

Does this means that an aircraft with flex wings will achieve a smoother ride under turbulence than an aircraft with non flex wings?

@Gabe Yes, it does.

Not only the 787 with CFRP has this, all wings flex a lot as shown by the lower part of this image.

Source: Introduction to Transonic Aerodynamics  by R. Vos and S. Farokhi
These days, designers incorporate the flexing into the design, making sure that the shape in cruise is exactly as they want it.
But the two graphs above show some interesting facts. On het left you can see the pressure distribution on different locations on a flexible wing, and on the right the same, but then for a rigid wing (thus, not deformed)
You can see that on the right image (around x/c=0.3), there are sharp jumps in the graphs, these indicate shocks, and lead to wave drag. On the flexible side, the gradients are less steep, meaning the shock wave is less strong. As a consequence the wave drag will be less.
Thus, based on these graphs, we can conclude that the flexible wing will have less wave drag, than the same wing that would not deform.

Good answer! But wouldn't the designers twist the wing just so that under load it has the desired angle of attack at all wing stations? After all, the result of flexing a backward-swept wing is to reduce the angle of attack at the outer stations. Of course the unflexed wing wing will have too much load at the outer stations.

That was also the thing I was aiming at. Designers know that the wing will deform, and will account for this in their design such that in cruise the shape is optimal. I made the comparison with the rigid case, not only to show that the flexing is a good thing, but also to explain why.